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BAIN MUSC 525

Post-Tonal Theory

Straus Ch. 1 Overview, Page 1 of 2

Straus Chapter 1 Basic Concepts and Definitions

Joseph N. Straus, Introduction to Post-Tonal Theory , 3rd ed. (Upper Saddle River, NJ: Prentice Hall, 2005).

“To appreciate [a] painting fully, you have to be willing to move from place to place.

One of the specially nice things about musicis that you can hear a single object like an interval in many ways at once.”

 – Joseph N. Straus, Introduction to Post-Tonal Theory

TERMS & CONCEPTS 

Twelve-tone equal temperament,

abbr. 12tet  

Semitone (1, the unit interval)

Equivalence

Equivalence relation1 

• Octave equivalence

• Enharmonic equivalenceEquivalence class

Pitch & Pitch-Class Notation

Pitch

• Frequency

• U.S. Standard pitch notation

(C4 = middle C)

• Staff notation

Pitch class, abbr. pc 

Letter notation

Integer notation

• Fixed-zero [C=0]

• Movable-zero: e.g. [A=0]

PC clockface diagram

Mod 12

• Modular arithmetic

• Modulus

• Congruence1 ! 11 + 2 (mod 12) 

Pitch Spaces

Linear pitch space, abbr. p-space 

Modular pitch-class space,

abbr. pc-space  

Intervals

Traditional tonal interval names

Ordered (melodic)

• Direction (signed, +/-)

• Magnitude (absolute value)

Unordered (harmonic)

Interval Types

1. Ordered pitch interval (opi)

2. Unordered pitch interval (upi)3. Ordered pc interval (opci)

4. Interval class (ic)

Interval Class

Interval class, abbr. ic

Straus ic scoreboard

Interval content

Interval vector

• Unique multiplicity of ic

Mod 12 complement

INTERVAL TYPES 

Space Interval Type Abbr.2  Values3  Definition

4  Description

 p- Ordered pitch interval (opi) ip -87 to 87 ip< x, y> = y- x  Melodic distance between two pitches 

Direction (+/-) and magnitude (in semitones)

Unordered pitch interval (upi) 0 to 87 ip( x, y) = | y- x| Harmonic distance between two pitches 

Magnitude only

 pc- Ordered pitch-class interval (opci) i 0 to 11  y- x (mod 12) Melodic distance between two pitch classes 

Clockwise distance (in semitones) on the

pc clockface diagram

INTERVAL CLASS (ic)

Unordered pitch-class interval (upci)

ic 0 to 6 The lesser of

 y- x (mod 12) and

 x-y (mod 12)

Harmonic distance between two pitch classes 

Shortest distance (in semitones) on the

pc clockface diagram

Straus says: “Which one we use will depend on what musical relationship we are trying to describe.”

EXAMPLE

 p-space pc-space

Pitch interval opi upi opci ic

A4 – Gƒ3  -13 13 11 1

1 For precise definitions of mathematical terms and concepts see MathWorld--A Wolfram Web Resource. Available online at:

<http://mathworld.wolfram.com>.2 Abbreviation used in Straus 2005 and throughout much of the atonal theory literature.3 For a linear p-space modeled by a 12tet piano keyboard (C4=0, with boundary conditions A0 = -39 & C8 = 48) and modular pc-space 

modeled by a pc clockface diagram.4 |x| is the absolute value of x.

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Post-Tonal Theory

Straus Ch. 1 Overview, Page 2 of 2

PITCH-SPACE VS. PITCH-CLASS SPACE 

In post-tonal theory, a 12tet pitch space ( p-space) is usually assumed. The 88 keys of the piano provide

a convenient reference for envisioning this linear space. The semitone has a magnitude of 1. It’s size

may alternatively be described as:

212

=21/12

 !

 1.059

 

P-space may be collapsed   into a pitch-class space  ( pc-space) under octave equivalence  (modeled by

mod 12). The pc clockface diagram provides a convenient reference for envisioning this modular space.

See Morris 1987, pp. 23-26 for more information.

QUOTABLE 

“[Pitch] is the intersubjective correlate of frequency.”

“Most musical systems quantize the pitch-continuum into a

finite number of pitches ordered from low to high.”

“[Relative pitch]...allows equal frequency ratios to be heard as equal pitch differences,

different intervals can be distinguished and grouped into classes.”

 – Robert Morris, Composition with Pitch Classes5 

“We locate pitches in an extended pitch space,

ranging in equal-tempered semitones from the lowest to the highest audible tone.”

“A broad consensus has emerged among music theorists regarding the

basic musical elements of post-tonal music–pitch, interval, motive, harmony, collection....”

“The quality of a sonority can be roughly summarized by listing all the intervals it contains.”

“Even though the interval-class vector is not as necessary a tool for tonal music as for post-tonal music,

it can offer an interesting perspective on traditional formations.”

 – Joseph N. Straus,  Introduction to Post-Tonal Theory

* * *

“Varèse’s octave is symmetrically partitioned by the tritone…

instead of asymmetrically by the fifth degree of the diatonic scale

into a perfect fifth and perfect fourth.”

 – George Perle, The Listening Composer6 

* * *

“I have chosen to define a good ear as one that perceives and retains musical structures

and understands their role in a musical transformation or other compositional process….

A good ear is an ideal toward which all musicians work in some fashion.”

 – Michael L. Friedmann, Ear Training for Twentieth-Century Music7 

5 Robert Morris, Composition with Pitch Class (New Haven: Yale University Press, 1987).6 George Perle, The Listening Composer (Berkeley: University of California Press, 1990).7 Michael L. Friedmann, Ear Training for Twentieth-Century Music (New Haven: Yale University Press, 1990).